Correlation Clustering and Biclustering with Locally Bounded Errors
This work addresses clustering with locally bounded errors, offering a more flexible approach for applications requiring balanced error distributions, though it is incremental in extending existing correlation clustering methods.
The paper tackles the correlation clustering problem by introducing a framework that generalizes the objective to functions of per-vertex errors, such as minimizing errors at the worst vertex, and provides a rounding algorithm that yields constant-factor approximations for various objective functions.
We consider a generalized version of the correlation clustering problem, defined as follows. Given a complete graph $G$ whose edges are labeled with $+$ or $-$, we wish to partition the graph into clusters while trying to avoid errors: $+$ edges between clusters or $-$ edges within clusters. Classically, one seeks to minimize the total number of such errors. We introduce a new framework that allows the objective to be a more general function of the number of errors at each vertex (for example, we may wish to minimize the number of errors at the worst vertex) and provide a rounding algorithm which converts "fractional clusterings" into discrete clusterings while causing only a constant-factor blowup in the number of errors at each vertex. This rounding algorithm yields constant-factor approximation algorithms for the discrete problem under a wide variety of objective functions.